TRANSFORMATIONS OF A LOGICAL PROPOSITION. 133 



and Y of Z, this does not, in any common or natural sense 

 of the word, constitute X a teacher of Z. 



The denial of these relations shall be indicated by r and 

 r~\ For our present purpose we shall use these symbols 

 with the following significations : — 



B 



teacher. 



£-> 



pupil. 



r 



not-teacher 



r -i 



not-pupil. 



These terms are to be understood in a purely qualitative 

 sense ; that is to say, the equation 



X=2?Y, 



and its inverse 



Y=i2- I X. 



mean respectively that X is a teacher of Y, and that Y is 

 a pupil of X, without implying whether or not X has any 

 other pupils and Y any other teachers. As in Boole's 

 notation, the coefficient i signifies all or every ; and I also 

 use the coefficient i —I , which is new — at least I have not 

 seen it used before. If 



X=iRY 



means that X is all the teacher of Y or, in ordinary lan- 

 guage, that X is the only teacher of Y, its inverse 



Y=i~ I i2- I X 



means that Y is the pupil of none but X, or of X only. 

 Thus, when standing before a relative term, i has the 

 meaning of only when used as an adjective, and i~ l has the 

 meaning of only when used as an adverb. The equation 



i-*=i, 



